A body of mass m kg starts falling from a point 2R above the earth's surface. Its kinetic energy when its has fallen to a point 'R' above the earth's surface [R Radius of earth M-mass of earth, G-gravitational constant]

Options

A. `1/2 (GMm)/R`
B. `1/6 (GMm)/R`
C. `2/3 (GMm)/R`
D. `1/3 (GMm)/R`

Explanation

To solve the problem, we need to determine the kinetic energy of a body of mass \( m \) when it has fallen to a height \( R \) above the Earth's surface, starting from a height \( 2R \). We will use the principle of conservation of mechanical energy, which states that the total mechanical energy (kinetic energy + potential energy) remains constant if only conservative forces (like gravity) are acting on the body.

### Step-by-Step Solution:

1. **Identify the initial and final positions:**
- Initial position: Height = \( 2R \) (above the Earth's surface)
- Final position: Height = \( R \) (above the Earth's surface)

2. **Calculate the initial potential energy (PE) at height \( 2R \):**
The formula for gravitational potential energy is given by:
\[ PE = -\frac{GMm}{r} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, \( m \) is the mass of the body, and \( r \) is the distance from the center of the Earth.

At height \( 2R \):
\[ PE_{initial} = -\frac{GMm}{R + 2R} = -\frac{GMm}{3R} \]

3. **Calculate the potential energy (PE) at height \( R \):**
At height \( R \):
\[ PE_{final} = -\frac{GMm}{R + R} = -\frac{GMm}{2R} \]

4. **Calculate the change in potential energy (ΔPE):**
The change in potential energy as the body falls from \( 2R \) to \( R \) is:
\[ \Delta PE = PE_{final} - PE_{initial} \]
Substituting the values:
\[ \Delta PE = \left(-\frac{GMm}{2R}\right) - \left(-\frac{GMm}{3R}\right) \]
\[ \Delta PE = -\frac{GMm}{2R} + \frac{GMm}{3R} \]
To combine these fractions, find a common denominator (which is \( 6R \)):
\[ \Delta PE = -\frac{3GMm}{6R} + \frac{2GMm}{6R} = -\frac{GMm}{6R} \]

5. **Using conservation of energy to find kinetic energy (KE):**
The change in kinetic energy is equal to the negative change in potential energy:
\[ KE = -\Delta PE = \frac{GMm}{6R} \]

### Final Answer:
The kinetic energy of the body when it has fallen to a height \( R \) above the Earth's surface is:
\[ KE = \frac{GMm}{6R} \]